二对边法向支承矩形边界平面问题新解法

A NEW METHOD FOR PLANE PROBLEM OF RECTANGULAR BOUNDARY WITH TWO OPPOSITE NORMAL SUPPORTED EDGES

新解法将平面问题分为广义静定和广义超静定二类,前者可以直接求解,后者要用叠加法求解。广义静定问题解由角点集中力解、体力分量解和计算边值条件解组成,这三种解要满足不同的变形协调条件,必须分别计算。角点集中力解、体力分量解与双调和方程无关,也不必满足边界条件;它们在边界处的应力和位移值反向作用在相应边界上为虚拟边值条件,实际边值条件和虚拟边值条件之和为计算边值条件。计算边值条件解即为经典的双调和方程解,由通解和特解组成。求解角点集中力解采用了隔离体平衡法。该文详细推导了二对边法向支承矩形边界平面问题所有解的表达式以及求解方法,并附有算例。

The plane problem can be divided into the generalized statically determinate and indeterminate problem, the former can be directly solved while the latter must be solved by the superposition principle. For the determinate problem, the final solution is composed of the corner concentrated force solution, the body force solution and the computed boundary-value condition solution. All of the three satisfy different strain compatibility equations and are determined separately. The first two solutions are independent of the bi-harmonic equation and boundary conditions, and the reverse stress and displacement of the two solutions at the four edges are the virtual boundary conditions. The sum of virtual and actual boundary conditions is the computed boundary-value condition, whose solution is the classical bi-harmonic equation solution and contains the homogeneous solution and particular solutions. The equilibrating of isolated free-body is used to determine the comer concentrated force solution. In this paper the expressions of all solutions and the evaluation process are presented for the plane problem of rectangular boundary with two opposite normal supported edges, and some examples are provided.